Decision making is a very vital process for every company and so as to come up with very good decisions, the company needs to adopt a very good method of calculating the probability of occurrence of some phenomenon. Some of the aspects which need to be considered when calculating probability include the highest probability of success of that particular event and the best fit i.e. the point that the company feels is good for their customers. Given the case, there are two methods which armed forces credit union will use to determine the probabilities. Because there are three options, each of them will present a possibility of one out of three. Thus the probabilities of any of the three options being picked are:

Cash back when the consumer makes an online purchase= 1/3

Cash back when the consumer makes a purchase at a shoe store= 1/3

Entry into a sweepstakes whenever the consumer makes a purchase= 1/3

For the first option, assumptions from the physical world will be used to make assumptions on the occurrence of the outcome, in this case the product development team assumes that there are 1,000 purchases and out of this only one purchase which will qualify for the award, then the probability will be calculated as 1/1000 (Davis et al., 2001). However, in a year, there is only an average of 52 purchases, which clearly shows that the possibility of a single purchase in a year is nil.

In this case the calculation of probability involves looking at the frequency of situations when they have been repeated over some time. The determination of probability therefore involves coming up with the number of times that the occurrence of the event is likely to occur. In this case since the occurrence is expected to be 52 within the year, then the probability will be 52/365, 365 indicating the number of days within the year (Davis et al., 2001).

These two probabilities are different in that they will not yield the same results. In the first option, the probability is nil. This sis because the only option is C, thus there is no likelihood of developing a plan that would beneficial to the organization. When determining the probability that there will be some purchases, it will be quite easy to use the second method than the first. These two methods are very appropriate in determining the probability because they give the long term and short term gains and losses which are attributed to the business. It implies that if the company wants to mitigate the risks which are associated with the occurrence of losses in the company, then it has to consider the giving prices at a constant rate (Hamburg, 2001). The third option is to have probability of an entry into a sweepstakes whenever the consumer makes a purchase. The probability of using the new card is 52/365. The probability that an individual customer will receive a prize over the course of the year is therefore calculated as follows:

1/3×1/1000×1/52= 1/156000

The probability that a person will receive a prize is one out of one hundred and fifty six thousand. Realistically, this may not be easily achievable given the number of people doing purchases. That is there are only 52 purchases in a year while, they target over a thousand purchases in one year. This is way far more than the average purchasing power.

Therefore the number of people that should be sampled should be representative of what is needed in an overall sampling size. Basically, an average number of card holders to be sampled should a figure representative and easy to deal with. Thus, 1000/52= 19.2. An average of 20 sample size would be appropriate.